Related papers: A construction of sequentially Cohen-Macaulay grap…
Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set $S$ is a vertex separator for vertices $a$ and $b$ if the removal of $S$ of the graph separates $a$ and $b$ into distinct connected components. A…
The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendleyet al. that for some rectangular grids, with…
We classify Cohen-Macaulay graphs of girth at least $5$ and planar Gorenstein graphs of girth at least $4$. Moreover, such graphs are also vertex decomposable.
Independence complexes of circle graphs are purely combinatorial objects. However, when constructed from some diagram of a link $L$, they reveal topological properties of $L$, more specifically, of its Khovanov homology. We analyze the…
We give a uniform and self-contained proof that if $G$ is a connected graph with $\chi(G) = \Delta(G)$ and $G\neq \overline{C_7}$, then $G$ contains either $K_{\Delta(G)}$ or an odd hole where every vertex has degree at least $\Delta(G)-1$…
The mapping class group of a closed surface of genus $g$ is an extension of the Torelli group by the symplectic group. This leads to two natural problems: (a) compute (stably) the symplectic decomposition of the lower central series of the…
To any simple graph \(G\), the clique graph operator \(K\) assigns the graph \(K(G)\) which is the intersection graph of the maximal complete subgraphs of \(G\). The iterated clique graphs are defined by \(K^{0}(G)=G\) and…
The biclique partition number $(\text{bp})$ of a graph $G$ is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the…
In 1966, Erd\H{o}s, Goodman, and P\'{o}sa showed that if $G$ is an $n$-vertex graph, then at most $\lfloor n^2/4 \rfloor$ cliques of $G$ are needed to cover the edges of $G$, and the bound is best possible as witnessed by the balanced…
Let $G$ be a finite graph and $I(G)$ its edge ideal. We give a full description of the Stanley--Reisner complex of the polarization of $I(G)^2$, naturally introducing the tools of Stanley--Reisner theory in the study of the algebraic…
The independence polynomial of a graph $G$ evaluated at $-1$, denoted here as $I(G;-1)$, has arisen in a variety of different areas of mathematics and theoretical physics as an object of interest. Engstr\"om used discrete Morse theory to…
We associate a {\it skew tableau ideal} to each filling of a skew Ferrers diagram with positive integers. We classify all unmixed and sequentially Cohen-Macaulay skew tableau ideals. Consequently, we classify all Cohen-Macaulay, Buchsbaum,…
A planar graph $G$ is said to be non-separating if there exists an embedding of $G$ in $\mathbb{R}^2$ such that for any cycle $\mathcal{C}\subset G$, all vertices of $G\setminus \mathcal{C}$ are within the same connected component of…
An {\em odd hole} in a graph is an induced subgraph which is a cycle of odd length at least five. An {\em odd parachute} is a graph obtained from an odd hole $H$ by adding a new edge $uv$ such that $x$ is adjacent to $u$ but not to $v$ for…
We study mixtures of decomposable graphical models, focusing on their ideals and dimensions. For mixtures of clique stars, we characterize the ideals in terms of ideals of mixtures of independence models. We also give a recursive formula…
Binomial edge ideals IG of a graph G were introduced by [4]. They found some classes of graphs G with the property that IG is a Cohen-Macaulay ideal. This might happen only for few classes of graphs. A certain generalization of being…
Applying a method of Godsil and McKay \cite{GM} to some graphs related to the symplectic graph, a series of new infinite families of strongly regular graphs with parameters…
Suppose that $F$ is a finite field and $R=M_n(F)$ is the ring of $n$-square matrices over $F$. Here we characterize when the Cayley graph of the additive group of $R$ with respect to the set of invertible elements of $R$, called the unitary…
The family of $\mathfrak{I}$-contractible graphs and contractible transformations was defined by A. Ivashchenko in the mid-90's. In this paper we study the collapsibility and homological properties of the clique complex associated to…
We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The…